Damping identification in multi-degree-of-freedom systems using the continuous wavelet transform

Abstract

The identification of damping in multi-degree-of-freedom vibration systems is a well-known problem and appears to be of crucial interest. Compared to an estimation of the stiffness and mass, the damping coefficient or, alternatively, damping ratio is the most difficult quantity to determine. In this paper, the continuous wavelet transform based on the Mor let-wavelet function is used to identify the modal damping ratios of multi-degree-of-freedom vibration systems. A new wavelet-based method for the damping identification frommeasured free responses is presented. The proposed method was also tested by experiments on a steel beam.

Full text

Vietnam Journal of Mechanics,
VAST,
Vol.
27,
No.
1 (2005), pp.
41
-
50
DAMPING
IDENTIFICATION
IN
MULTI-DEGREE-OF-FREEDOM
SYSTEMS
USING
THE
CONTINUOUS
WAVELET
TRANSFORM
NGUYEN
PHONG
DIEN
Hanoi University
of
Technology
A
bs
tr
ac
t . The identification of damping in multi-degree-of-freedom vibration systems
is
a
we
ll-
known probl
em
and appears to be
of
crucial interest. Compared to an estimati
on
of
the stiffness and mass, the damping coefficient or, alternativel
y,
dampi
ng
ratio
is
the most
difficult quantity to determine.
In
th
is paper, the continuous
wave
let transform based on
the
Mor
let-wavelet function
is
used to
id
enti
fy
the modal damping ratios of multi-degree-of-
freedom vibration systems. A
new
wave
let-based method
for
the damping identification
from
measured
free
responses
is
presented. The proposed method
was
al
so
tested
by
experiments
on a steel beam.
1.
IN
T
ROD
UCT
ION
The
identificati
on
of
damping
in multi-degree-of-freedom
vibration
systems is a well-
known
prob
lem
and
appears
to
be of crucial interest.
Compared
to
an
estimation
of
the
stiffness
and
mass,
the
damping
coefficient or, alternatively,
damping
ratio
is
the
most
difficult
quantity
to
determine. While
both
mass
and
stiffness
can
be
determined
by
static
tests,
damping
requires a
dynamic
test
to
measure.
The
major
i
ty
of
damping
measurements
performed
today
are
based
on
experimental
modal
ana
lysis or
modal
testing
[8]
. However,
this
approach
requires a specific
measurement
hardware
and
a complex
software for
determining
the
frequency response function of
the
system
and
extracting
the
moda
l
data.
Furthermore,
the
frequency response functions will often give significant
errors resul
ting
from
the
influence of
the
noise
and
the
spectrum
overlap.
Over t he
past
10 years, wavelet
theory
has become one of
the
emerging
and
fast-
evolving
mathematical
and
signal processing
tool
for
its
many
distinct
mer
i
ts
. General
overview of
app
li
cation
of
the
wavelet
transform
may
be
found in
[2,
6,
7].
The
continuous
wavel
et
transform
has been proved
to
have a high level of efficiency
in
accurate
information
and
short
processing
time
.
The
wavelet
transform
has
dominant
advantages
in
si
gna
l
fi
l
tering
and
ot
h
er
merits
such as
time
-frequency characteristics, which
make
it possible
to
parameter
identification [3-5].
In
this
paper,
the
continuous wavelet
transform
based
on
the
Morlet-wavelet function
is used
to
identify
the
moda
l
damping
ratios
from free responses of linear
vibration
systems
wit
h multi-degree-of-freedom.
The
following examples
demonstrate
the
improvemen
ts
of
the
proposed wavel
et
-
based
method
on
the
damping
identification.
2.
THE
CON
T
INUO
US WAV ELET
TR
AN
SFORM
This
section presents a brief background
on
the
Continuous Wavel
et
Transform
(CWT)
ut
ilized
in
th
is paper. A
thorough
discussion of wavelets is given
in
references
[1],
[7
].

42
Nguy
en
Phong
Dien
Using a
mother
-wavelet function
'1/i
(t), wavelets
are
family of functions of
type
1
(t
-
T)
'l/!T,s(t)
=
..fi
'l/i
-8-
l T , s
ER+
'
(2
.
1)
generated from
'lji
(t) by
the
operation
of
dilation
bys
and
translation
by
T.
In
the
time
domain,
'l/!
7
,s
is centered
at
T
with
a spread
proportional
to
s.
The
CWT
of a signal x(t)
is defined by:
1
/+oo
(t
-
T)
CWT
{x(t)}
=
Wx(T
, s) =
Vs
x(t)'lji* - 8
-dt, (2.2)
-oo
where
'lji
* is
the
complex conjugate of
the
wavelet
'lji
. A wavelet coefficient
Wx(T
,s) mea-
sures
the
variation of signal x(t) in a neighborhood of position T .
By
varying
the
scale s
and
translating
along
the
localized
time
T,
the
amplitude
of wavelet coefficients
IW
x
(T
, s)I
can
construct
a wavelet amplitude map showing
both
the
amplitude
of
any
features in
the
signal versus
the
scale
and
how
this
amplitude
varies
with
time
. Therefore, information
in
the
time
domain
will still remain, in
contrast
to
the
Fourier Transform
(FT),
where
the
time
domain
information becomes almost invisible
after
the
integration
over entire record
length
of
the
signal.
+oo
FT
{x(t)} = x(w) = J x(t)eiwtdt. (2.3)
-
oo
A complex mother-wavelet
can
be
constructed
with
a frequency
modulation
of a real
and
symmetric
window g(t).
'l/i
(t) = g(t)
ei
71ot,
(2.4)
where
T/o
is a
constant
parameter
and
i -
.J=I.
A Morlet mother-wavelet is
obtained
with
a Gaussian window. g(t) =
7r
- l /
4e
- t2/ 2
' (2.5a)
g(w)
=
7r-l/4
e - w2 / 2 (2.5b)
In
this
study
the
Morlet wavelet
is
used
to
perform
the
CWT.
The
relation between
the
scale s
and
the
frequency w of
the
Morlet wavelet
can
be derived analytically as
[1
J.
s =
T/O
w
(2
.
6)
As
can
be
seen from
(2
.2),
the
CWT
is a linear time-frequency transform.
Its
linearity
makes
it
possible
to
analyze each i-th component Xi(t) of a multi-component signal.
CWT
{
t,
x;
(t)} =
t,
CWT
{x;(t)} (2.7)

Damping
Identification
in
Multi-Degree-of-Fr
ee
dom
Systems
...
43
3.
MODAL
DAMPING
RATIO
IDENTIFICATION
3.1
Signal
envelope
extraction
with
the
CWT
Consider a signal x(t) th
at
can
be
expressed by a
modulated
sinusoidal function
x(t) = a(t) cos(aot +
/3)
= a(t) cos<p(t) , (3.1)
where a(t) is
the
envelope of x(t)
and
<p(t)
= aot +
/3
.
The
CWT
of
this
signal
can
be
derived
analytica
lly by a simple function as follows
(see also
[1])
(3
.
2)
where
g(rJo
-sa0) is
the
Fourier
transform
of g(t)
at
w =
'f"/o
-
sao
and
c(T,
s)
is
the
corrective
term
.
If
the
variations of a(t) are slow compared
to
the
period
27r
/
ao,
the
term
c(T
,
s)
can
be
neglected [1,3].
In
this
case, by considering expression
(2
.6
)
the
CWT
of
x(t)
can
be
written
in
the
form:
(3.3)
The
amplitude
of wavelet coefficients
IWx(T,
s)I
is given by
(3.4)
Since
1.§(w)I
is
maximum
at
w =
0,
expressi~n
(3.4) shows
that
IWx(T,w)I is
max
imum
at
w = a0. For w = 0 expression
(2
.5b) becomes §(0) =
7r
-1/4.
The
envelope a(T)
at
w =
ao
is given by (3.5)
where K is a positive
constant
K =
27r
114J
ao/'f"Jo
.
Expression (3.5) shows
that
the
wavelet coefficients IWx
(T
,
w)I
at
w =
ao
(called as
wav
elet envelope
[5])
is
proportional
to
the
envelope of signal x(t) described in (3. 1).
3.2
Damping
ratio
estimation
using
CWT
It
is well known
that
a
damped
n -degree-
of
-freedom
system
with
proportional
damp-
ing
Mq
+ Bq +
Cq
=
0,
(3.6)
can
be
solved by using
the
modal
ana
lysis
[8].
This
leads
on
n decoupled modal
equat
ions
(3
.7
)
Here Pi(t) denotes
th
e i-
th
modal coordinate,
wo
i is
the
i- th
und
amped
natural
frequency
and
(i is
the
i-
th
modal
damping
ratio. T he
modal
damping
places
an
energy
dissipation
term
of
the
form 2(iwoiJ\(t) (viscous damping).
This
form is chosen largely
for its
mathematical
convenience.
The
modal
damping
ratios (i are assigned by making
measurements of
the
free
damped
response
and
estimating
(i·

44 Nguyen Phong Dien
The
task
of
interest
is
to
determine
the
modal
damping
ratios associated
with
each
mode
shape
.
The
linearity of
the
CWT
is
useful for
extracting
modal
data
of each mode
from measured free responses of
the
system
.
This
will be
demonstrated
in section
4.
Consider
the
free response of a linear
vibration
system
for
the
unterdamped
case,
the
signal model corresponding
to
the
i-th
mode can be described by
xi (t) = Aie-(;wo; t sin(wdi t +
</>i),
(3.8)
where
amplitude
A is a
constant
and
Wdi
is
the
i-
th
damped
natural
frequency
The
decay
envelope of
this
signal is ai(t) = Aie- (;
wo;t.
Note
that
it
is difficult
to
extract
the
envelope
ai(t) from measured free response
with
numerous
natural
frequencies. According
to
the
signal model
(3
.8),
the
logarithmic decrement
8i
is given by
8i
=
_!_
ln ai(t)
m ai(t +
mTi)'
(3.9)
where m is any positive integer
and
Ti =
27r
/wdi· Since
the
variations of ai(t) are slow
compared
to
the
period
Ti
, using expressions (3.5)
and
(3
.
9)
the
logarithmic decrement
8i
can
be
expressed in
term
of wavelet coefficients as
J:
_
_!_
l JWxi(t,
WdiJ
ui -n )I.
m JWxi(t + mTi,
Wdi
(3.10)
Finally,
the
modal
damping
ratio
can be determined from logarithmic decrement
8i
~·
.
( -
8i
. -
V47r2
+of (3.11)
In
summary,
the
following procedure is required for identifying
damping
ratios:
-Detecting local
maxima
in
the
wavelet
amplitude
map
to
determine
natural
frequen-
cies.
-
Extracting
the
wavelet envelope IWx(T,w)J
at
th_e
natural
frequencies.
-Calculating
the
damping
ratios
(Formulas (3.10)
and
(3
.11)).
A specialized
program
has been developed
on
the
MATLAB®
numeric
computing
environment for
this
study
.
The
program
includes two modules:
-Signal processing functions for
the
continuous Wavelet-Transform.
-Calculating
the
damping
ratios according
to
the
above-mentioned procedure.
4.
NUMERICAL
EXAMPLE
In
order
to
assess
the
performance of
the
proposed wavelet-based
method,
a
test-
signal
with
three
exponentially-decaying components has been chosen for
simulating
the
free response of a
damped
vibration
system.
3
x(t) = L aie-
(;wo;t
sin( wait+
</>i)
·
i=l
(4.1)

Damping Identification in Mul
ti
-D
egree
-
of
-Preedom
Syst
e
ms
.
..
45
Table
1.
Parameters
of
the
test
-signal
Component
i 1 2 3
fi =
Wdi
/ 27r(Hz) 250 500 1250
(i
0.02 0.045 0.005
<Pi
-7f
/8
7f
/6
7f
/8
ai
1.0 1.25 0.5
Fig. 1 shows
the
signal
in
the
time
domain.
The
parameters
of
the
si
gna
l
are
given
in
Table 1.
Th
e fre
qu
encies of signal-components
are
differe
nt
, the l
ast
component has a
high frequency
with
a
sma
ll
damping
ratio.
-1
-2'--~~~~~~
~~~
~~~~~~~~~
4
...
2
0
2000
1500
Frequency (Hz)
0 0.
05
0 1 0.
15
0.2 (s)
1000
Fig
.
1.
Th
e test-si
gn
al
3.
Component
2.
Component · · · ·
l
0.1
0
1. Component
0.15
T
im
e 1 (s)
0.2
Fig
. 2.
The
wavelet a
mplitude
map
of
the
test-signal
Firstly,
the
signal is
tr
ansformed
in
time
-frequency
domain
using
the
CWT
with
Morlet-wavelet.
Fig
. 2 is the wavelet
amp
li
tude
map
displayed as a three-dimensional
s
urfa
ce,
obtained
by
plotting
the
amp
li
tude
of
the
wavelet coeffici
ents
JW
x
(T
,
w)
J
aga
inst

46
Nguy
en
Phong
D
ien
time
a
nd
fr
eq
uency.
Three
expon
ent
ially
-d
ecaying components
are
separated
from
the
original signal
an
d
ca
n be clearly ident
ifi
ed
.
In
o
rd
er
to
ident ify
"natura
l" fr
eq
uency
wai
of each
signa
l-compon
ent
from
the
map
in
Fig
. 2, a numerical algori
thm
is developed
to
seek l
oca
l
maxima
of t he three-dimensional
surface
[4].
The
positions of
the
l
oca
l
maxima
reveal the correspo
ndin
g
natural
frequencies
in
frequency
ax
is (see secti
on
3.1).
The
wavel
et
envelope I
Wx
( T,
wai)
I
ca
n
be
extracted
from
the
wavelet a
mplitud
e map
by a slice
parall
el
to
th
e t ime axis t
hrou
gh
frequency
wai
in
the
frequency
ax
is. According
to
formulas (3.10)
and
(3.
11
),
this
envelope
ca
n be used
to
determine
the
damping
ratio
of signal components.
Fig
. 3 shows
the
envelope
extracted
at
the
th
i
rd
natural
frequency.
•ll
<:l
::::l
-
Q.
:=
1.5-~-----r--,--.--~-~-r--r-----r-:-i
'
------
~
------: ---
r-
-\
-----: ------: ------: ------: ------: ------: ------:
' ' '
'
4':
0.5 ------
~
------
~
--f ---
~
----
-x-
-----
~
------
~
------
~
------: ------: -----.-:
o~-..JL..---L..l_-1._-1.
_
___L
_
___c:::=~~~=--...1....--..L..J
0 0.
02
0
04
0
06
0
08
0 1 0.12 0.
14
0.
16
0.
18
0.2 t (s)
Fig.
3.
The
wave
let
enve
lope of the
th
ird signa
l-
component extracted
from
the map in Fi
g.
2
Table 2. R
es
ults of damping identification
for
the test-signal
Paramet
er i = 1 i = 2
i=3
wai/27r(Hz)
247
500
1255
Ti
(s)
4x
10
-0
2x
10
-0 7
.9
7x
10
-4
m
10
15
50
TO
0.080 0.0567 0.060
IWx(
To,wa
i)I
2.6167 2.5620 0.9835
IWx(
To+ m'li,
wai)
I 0.7340 0.0485 0.2060
c5
i 0.1271 0.2645 0.0313
(i
0.0202
0.0421
0.005
Error
of
W(Ji
(%)
1.1 0.0
0.
4
Error
of
(i
(%)
1.0 6.4 0.0
Th
e
id
ent
ifi
cat
i
on
data
is given in Table
2.
It
ca
n be seen
that
the
id
entification of
the
natural
frequencies
and
the
damping
ratios
is very good.
5.
EXPERIMENT
Analysis of
the
measured
impul
se response of a
beam
provides a good de
monstration
of
t'
he proposed wavelet-based m
etho
d.
Th
e experiment was ·
don
e
at
a uniform steel
beam
of

Damping Identification
in
Multi-Degree-
of
-Freedom
Systems
.
..
47
rectangular cross-section
(25x30mm)
with
the
clamped-clamped bo
und
ary config
ura
tion
(
Fi
g.4).
data-acquisition system sensor steel beam
Fig.4.
Exper
i
mental
set
-u
p
(m/s2)
40
.----.---
--.--------.---~---
---,-
---
--,
' ' .
------- - - - , ----------
-,
-- - - - - - -- - -.------ - - - -
20
0
. .
-20 I I I I
- - - T - - - - - - - - - - , - - - - - - - - - -
-i
-- - - - - - - - -
-,-
- - - - - - - - -
Fig.
5.
Measured
vibration
response of
the
beam:
time
record
(m/s2) 3
.----
--~--
--~-----~------,
2044 Hz
2.5
2
1.5 706
Hz
1266 Hz
0.5
I 3066
Hz
oi.:=:::::::::==:::J...====-'A~==~~JL\....~="'=~--~_A~~~~_J
0
1000
2
000
3
000
4000
(Hz)
Fig. 6. Measured
vibration
response of
the
beam
: frequency
spectrum

48 Nguy en Phong Dien
Wavelet amplitude
·:··
100 "" 3. mode 2. mode
50
0
4000
·· · 4. mode
//
1m00e
·.
Time (s)
0.1 0.2
Frequency
0
Fig.
7.
Th
e wavelet
amplitud
e map of
th
e
mea
s
ur
ed
vibration
signal shown
in
Fig
. 5
35.-~~~-,--~~~----,-~~~~-,----~~~~~~~-----,
30
25
0)
-g
20
E-
15
·
~
10
- -
---J-----
-
------L--------
-
--J
-
-----
-
--
-
--L---
-
-------
1 I I I
' . ' '
-----~-------
-
----r-
-
---------~----
- -
-
-----r-----------
' I I I
-
--,------------r-----
-
-----,--------
-
---r-----------
I I I I
----~--,-----------
-
r---------
-
-
,
-
-----
-
-----r
-
-----
- - -
--
5
00
---------~----------
-
~------------~-----------
0 .1
0.2
11
-.
- . ,
.J
0.4
Fig. 8.
Th
e
wa
velet envelope at
th
e 2nd na
tur
al
fr
e
qu
ency
70
60
-----~---
-
--------~
-
----
-
-----~---------
-
--~-----------
'
I I I I
50
----
-
-~------------~-----------
~
---------
-
--~--
-
--------
0)
-g
411
·= -
-------~------------~-
-
-----
-
-
--
~--
-
--------
-
~-----------
E
30
-------~------
-
-----~-----------~------------~-----------
4'.
20
-------
~
-
_,_
--------
~
------ - ----
...:
------------
~
-------- - - -
..
10
0 0 0.1
02
0 .3
04
Fig.
9.
Th
e wavelet envelope
at
th
e 3
rd
na
tural
frequency

Damping
Id
entifica
tion
in
Multi
-D
egree-of-Preedom
Systems
... 49
A
hammer
with
a
hard
tip
and
a soft
tip
was
used
for
generat
in
g
impacts
.
An
accelerometer
on
the
beam
at
the
closest
position
to
the
point
of
the
impact
measured
response of
bending
v
ib
rat
ion
of
the
beam.
The
vibration
signa
ls were
sampled
at
the
samp
ling frequency
of
10
kHz by a
multi-channel
data-acquisition
system.
The
time
record
of
the
vibration
response is
shown
in
Fig. 5.
The
natural
frequencies
of
the
beam
can
be
determined
in
advance
by
means
of
the
spectra
l analysis
with
FFT
.
Fig
. 6 displays
the
spectrum
with
the
first four
natural
fre
quen
cies.
Since
the
acceleration
of
the
free response
at
each
vibration
mode
is
nearly
proportional
to
the
corresponding
displacement
[3]
,
the
measured
acce
leration
signal
can
be
directly
used
for
damping
id
enti
fic
at
ion. Fig. 7 displays
the
wavelet
amplitude
map
of
the
measured
signal.
Th
e wavelet envelopes displayed
in
Figs. 8
and
9 were
used
for
estimating
th
e
damping
ratios.
Table
3.
Results of damping identification
Natural
frequencies
[Hz]
Mode
FFT-method
CWT-method
(
Mode
1 706 707 0.0015
Mod
e 2 1266 1269 0.0032
Mode
3 2044 2044 0.0012
Mode
4 3066 3084 0.0041
The
main
results
of
the
damping
identification for
the
beam
are
given in
Tabl
e 3.
Not
e
that
the
damping
ratio
(i
of
the
bending
vibration
of
steel is
small
unl
ess
the
structure
contains
viscoelastic
material
or
a
hydraulic
damper
is present.
Common
values
are
ex-
pected
to
be
in
the
range
0
:::;;
(i
:::;;
0.05 (reference [8]). Therefore,
the
results
seem
to
be
in
good
agreement.
6.
CONCLUSIONS
A wavelet-based
method
for
the
damping
identification from
measured
free response
of
multi-degre
e-of-fr
eedom
vibration
systems
is
presented
.
The
proposed
method
carries
out
the
continuous
wavelet
transform
of
the
response
and
the
extract
ion
wavelet envelopes
in
time
-fr
eq
uency
domain.
The
damping
ratio
of
each
vibration
mode
is
then
d
eter
mined
by
the
value
of
the
lo
gar
ithmic
decrement
.
The
wavelet
-b
ased
method
was also
tested
by
experiments
on
a
steel
beam.
As
can
be
seen from t
he
identification
data,
th
is innovative
approach
proves
to
be
a very effective
signal processing
tool
for
the
ex
perim
enta
l
vibration
ana
lysis.
Acknowledgment
.
This
paper
was
comp
leted
with
the
finanGial
support
of
the
Vietnam
Basic Res
ea
rch
Pro
gram
in
Natural
Science.
REFERENCES
l.
S.
Mallat , A Wav elet Tour
of
Signal Processing, Academic Press, San
Di
ego,
London,
New
York, 1999.
2.
Z.
K.
Peng , F. L. Chu , Application of the wavelet transform in machine condition monitoring
and fault diagnostics: a review with bibliography, Mechanical Systems and Signal Processing
18
(2004) 199-221.

50 Nguyen Phong
Dien
3. J. Slavic,
I.
Simonovski, M. Boltezar,
Damping
identification using a continuous wavelet
transform:
app
li
cation
to
real
data.
Journal
of
Sound and Vibration
262
(2003) 291-307.
4.
D. E. Newland, Ridge
and
Phase
Identification
in
the
Frequency Analysis of Transient Signals
by Harmonic Wavelets,
ASME
Journal
of
Vibration and Acoustics
121
(1999) 149-155.
5.
E.
L.
Schukin,
R.
U. Zamaraev,
L.
I. Schukin,
The
optimisation
of wavelet
transform
for
the
impulse
ana
lysis in
vibration
signals. Mechanical Syste
ms
and Signal Processing
18
(2004)
1315-1333.
6.
G. Meltzer, Nguyen
Phong
Dien,
Fault
diagnosis in gears
operating
under
non-stationary
ro-
tational
speed using
polar
wavel
et
amplitude
maps
. Mechanical Systems and Signal Processing
18
(2004) 985-992.
7. Nguyen
Phong
Dien, Beitrag Zur Diagnostik Der Verzahnungen
in
Getrieben Mittels Zeit-
Frequenz-Analyse, Fortschritt-Berichte VDI, Reihe
11
, Nr. 135, VDI-Verlag, Dueselldorf, 2003.
8.
D. J.
Inman
, Engin
ee
ring Vibration, Prentice-Hall
Publisher
, New Jersey, 2001.
Received September 6, 2004
Revised November
24
, 2004
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